Finite Element Modeling of Electromagnetic Systems

Transcription

1 Finite Element Modeling of Electromagnetic Systems Mathematical and numerical tools Unit of Applied and Computational Electromagnetics (ACE) Dept. of Electrical Engineering - University of Liège - Belgium Patrick Dular, Christophe Geuzaine February 215 1

2 Introduction v Formulations of electromagnetic problems Maxwell equations, material relations Electrostatics, electrokinetics, magnetostatics, magnetodynamics Strong and weak formulations Discretization of electromagnetic problems Finite elements, mesh, constraints Very rich content of weak finite element formulations 2

3 Formulations of Electromagnetic Problems Electrostatics Maxwell equations Electrokinetics Magnetostatics Magnetodynamics 3

4 Electromagnetic models All phenomena are described by Maxwell equations Electrostatics Distribution of electric field due to static charges and/or levels of electric potential Electrokinetics Distribution of static electric current in conductors Electrodynamics Distribution of electric field and electric current in materials (insulating and conducting) Magnetostatics Distribution of static magnetic field due to magnets and continuous currents Magnetodynamics Distribution of magnetic field and eddy current due to moving magnets and time variable currents Wave propagation Propagation of electromagnetic fields 4

5 Maxwell equations Maxwell equations curl h = j + t d curl e = t b div b = div d = ρ v Ampère equation Faraday equation Conservation equations Principles of electromagnetism Physical fields and sources h magnetic field (A/m) e electric field (V/m) b magnetic flux density (T) d electric flux density (C/m 2 ) j current density (A/m 2 ) ρ v charge density (C/m 3 ) 5

6 Material constitutive relations Constitutive relations b = µ h (+ b s ) d = ε e (+ d s ) j = σ e (+ j s ) Magnetic relation Dielectric relation Ohm law Characteristics of materials µ magnetic permeability (H/m) ε dielectric permittivity (F/m) σ electric conductivity (Ω 1 m 1 ) Possible sources b s remnant induction,... d s... j s source current in stranded inductor,... Constants (linear relations) Functions of the fields (nonlinear materials) Tensors (anisotropic materials) 6

7 Electrostatics Basis equations curl e = div d = ρ d = ε e & boundary conditions n e Γ e = n d Γ d = e electric field (V/m) d electric flux density (C/m 2 ) ρ electric charge density (C/m 3 ) ε dielectric permittivity (F/m) Type of electrostatic structure Electric scalar potential formulation div ε grad v = ρ with e = grad v Formulation for the exterior region Ω the dielectric regions Ω d,j In each conducting region Ω c,i : v = v i v = v i on Γ c,i Ω Exterior region Ω c,i Conductors Ω d,j Dielectric 7

8 Electrokinetics Basis equations curl e = div j = j = σ e & boundary conditions n e Γ e = n j Γ j = e electric field (V/m) j electric current density (C/m 2 ) σ electric conductivity (Ω 1 m 1 ) Type of electrokinetic structure Γ j Γ e,1 Ω c Γ e, e=?, j=? Electric scalar potential formulation div σ grad v = with e = grad v V = v 1 v Ω c Conducting region Formulation for the conducting region Ω c On each electrode Γ e,i : v = v i v = v i on Γ e,i 8

9 Electrostatic problem Basis equations curl e = d = ε e div d = ρ S e S e 1 S e 2 F e F e 1 F e 2 F e 3 grad curl div e e e ( v ) e ε e = d ρ d ( u ) div curl d grad d d F d 3 F d 2 F d 1 F d S d 3 S d 2 S d 1 S e 3 "e" side e = grad v d = curl u "d" side S d 9

10 Electrokinetic problem Basis equations curl e = div j = j = σ e S e S e 1 S e 2 F e F e 1 F e 2 F e 3 grad curl div e e e ( v ) e σ e = j j ( t ) div curl j grad j j F j 3 F j 2 F j 1 F j S j 3 S j 2 S j 1 S e 3 "e" side e = grad v j = curl t "j" side S j 1

11 Classical and weak formulations Partial differential problem Classical formulation L u = f in Ω B u = g on Γ = Ω u classical solution Weak formulation Notations ( u, v ) = u ( x ) v ( x ) d x, u, v L 2 ( Ω ) Ω ( u, v ) = u ( x ). v ( x ) d x, u, v L 2 ( Ω ) Ω ( u, L * v ) - ( f, v ) + Q g ( v ) ds =, v V ( Ω ) u weak solution Γ v test function Continuous level : system Discrete level : n n system numerical solution 11

12 Classical and weak formulations Application to the magnetostatic problem curl e = div d = Γ e d = ε e Γ d n e Γe = n. d Γd = Electrostatic classical formulation ( d, grad v' ) =, v' V(Ω) with V(Ω) = { v H (Ω) ; v Γe = } ( div d, v' ) + < n. d, v' > Γ =, v' V(Ω) div d = n. d Γd = Weak formulation of div d = (+ boundary condition) d = ε e & e = grad v curl e = ( ε grad v, grad v' ) =, v' V(Ω) Electrostatic weak formulation with v 12

13 Classical and weak formulations Application to the magnetostatic problem curl h = j div b = Γ h b = µ h Γ e n h Γh = n. b Γe = Magnetostatic classical formulation ( b, grad φ' ) =, φ' Φ(Ω) with Φ(Ω) = { φ H (Ω) ; φ Γh = } ( div b, φ' ) + < n. b, φ' > Γ =, φ' Φ(Ω) div b = n. b Γe = Weak formulation of div b = (+ boundary condition) b = µ h & h = h s grad φ (with curl h s = j) curl h = j ( µ (h s grad φ), grad φ' ) =, φ' Φ(Ω) Magnetostatic weak formulation with φ 13

14 Quasi-stationary approximation curl h = j + t d Dimensions << wavelength Conduction current density > > > Displacement current density Applications curl h = j Electrotechnic apparatus (motors, transformers,...) Frequencies from Hz to a few 1 khz 14

15 Magnetostatics Equations curl h = j Ampère equation Ω Type of studied configuration div b = Magnetic conservation equation Ω m Ω s j Constitutive relations b = µ h + b s j = j s Magnetic relation Ohm law & source current Ω Studied domain Ω m Magnetic domain Ω s Inductor 15

16 Magnetodynamics Equations curl h = j Ampère equation Ω Type of studied configuration Ω s curl e = t b div b = Faraday equation Magnetic conservation equation Ω p j s I a Constitutive relations b = µ h + b s j = σ e + j s Magnetic relation Ohm law & source current Ω a Ω Studied domain Ω p Passive conductor and/or magnetic domain Ω a Active conductor Ω s Inductor V a 16

17 Magnetic constitutive relation b = µ h µ = µ r µ µ r relative magnetic permeability Diamagnetic and paramagnetic materials Linear material µ r 1 (silver, copper, aluminium) Ferromagnetic materials Nonlinear material µ r >> 1, µ r = µ r (h) (steel, iron) b-h characteristic of steel µ r -h characteristic of steel 17

18 Magnetostatic formulations Ω Ω m Ω s j Maxwell equations (magnetic - static) curl h = j div b = b = µ h φ Formulation a Formulation "h" side "b" side 18

19 Magnetostatic formulations Basis equations curl h = j b = µ h div b = (h) (m) (b) φ Formulation Magnetic scalar potential φ h = h s grad φ Multivalued potential h s given such as curl h s = j (non-unique) div ( µ ( h s grad φ ) ) = Cuts (h) OK (b) & (m) a Formulation Magnetic vector potential a (h) & (m) (b) OK b = curl a curl ( µ 1 curl a ) = j Non-unique potential Gauge condition 19

20 Multivalued scalar potential Kernel of the curl (in a domain Ω) ker ( curl ) = { v : curl v = } grad dom(grad) cod(grad) curl ker(curl) dom(curl) curl. cod(curl) cod ( grad ) ker ( curl ) 2

21 Multivalued scalar potential - Cut curl h = in Ω OK? Scalar potential φ h = grad φ in Ω Circulation of h along path γ AB in Ω h. d l = - grad φ. d l = φ A - φ B γ AB γ AB I γ Closed path γ AB (A B) surrounding a conductor (with current I) φ A φ B = I!!! (Σ) Ω φ must be discontinuous... through a cut Δ φ = I 21

22 Vector potential - gauge condition div b = in Ω OK? Vector potential a b = curl a in Ω Non-uniqueness of vector potential a b = curl a = curl ( a + grad η ) Gauge condition ex.: w(r)=r Q Coulomb gauge div a = O Gauge a. ω = ω vector field with non-closed lines linking any 2 points in Ω P 22

23 Magnetodynamic formulations Ω Ω s h-φ Formulation Ω p j s I a a * Formulation Ω a V a t-ω Formulation Maxwell equations (quasi-stationary) curl h = j curl e = t b div b = b = µ h j = σ e a-v Formulation "h" side "b" side 23

24 Magnetodynamic formulations curl h = j (h) Basis equations b = µ h j = σ e curl e = t b div b = (b) h-φ Formulation Magnetic field h Magnetic scalar potential φ h ds Ω c h = h s grad φ ds Ω c C curl (σ 1 curl h) + t (µ h) = div (µ (h s grad φ)) = (h) OK curl h s = j s (b) in Ω c in Ω c C t-ω Formulation Electric vector potential t Magnetic scalar potential ω (h) OK curl (σ 1 curl t) + t (µ (t grad ω)) = div (µ (t grad ω)) = j = curl t h = t grad ω + Gauge 24

25 Magnetodynamic formulations curl h = j (h) Basis equations b = µ h j = σ e curl e = t b div b = (b) a * Formulation Magnetic vector potential a * a-v Formulation Magnetic vector potential a Electric scalar potential v b = curl a * e = t a * (b) OK (b) OK b = curl a e = t a grad v curl (µ 1 curl a * ) + σ t a * = j s (h) curl (µ 1 curl a) + σ ( t a + grad v)) = j s + Gauge in Ω c C + Gauge in Ω 25

26 Magnetostatic problem Basis equations curl h = j div b = b = µ h S h S h 1 S h 2 F h 1 F h 2 F h 3 F h grad curl div h h h (œφ) h j µ h = b b (a) div curl grad 3 F e e 2 F e e F e 1 e F e S e 3 S e 2 S e 1 S h 3 "h" side h = grad φ b = curl a "b" side S e 26

27 Magnetodynamic problem curl h = j Basis equations b = µ h j = σ e curl e = t b div b = S h S h 1 S h 2 F h 1 F h 2 F h 3 F h grad curl div h h h (œφ) (t) h j µ h = b j = σ e div e b curl e (a, a *) grad (œv) e 3 F e 2 F e 1 F e e F e S e 3 S e 2 S e 1 S h 3 "h" side h = t grad φ b = curl a e = t a grad v "b" side S e 27

28 Continuous mathematical structure Basis structure Domain Ω, Boundary Ω = Γ h U Γ e Function spaces F h L 2, F h 1 L 2, F h 2 L 2, F h 3 L 2 dom (grad h ) = F h = { φ L 2 (Ω) ; grad φ L 2 (Ω), φ Γh = } dom (curl h ) = F h 1 = { h L 2 (Ω) ; curl h L 2 (Ω), n h Γh = } dom (div h ) = F h 2 = { j L 2 (Ω) ; div j L 2 (Ω), n. j Γh = } grad h F h F h 1, curl h F h 1 F h 2, div h F h 2 F h 3 Sequence F h grad h Boundary conditions on Γ h curl h div h F h F h F h Basis structure Function spaces F e L 2, F e 1 L 2, F e 2 L 2, F e 3 L 2 dom (grad e ) = F e = { v L 2 (Ω) ; grad v L 2 (Ω), v Γe = } dom (curl e ) = F e 1 = { a L 2 (Ω) ; curl a L 2 (Ω), n a Γe = } dom (div e ) = F e 2 = { b L 2 (Ω) ; div b L 2 (Ω), n. b Γe = } grad h F e F 1 e, curl e F 1 e F 2 e, div e F 2 e F 3 e Boundary conditions on Γ e Sequence F e 3 div e 2 curl F e 1 grad e F e e F e 28

29 Discretization of Electromagnetic Problems Nodal, edge, face and volume finite elements 29

30 Discrete mathematical structure Continuous problem Continuous function spaces & domain Classical and weak formulations Discretization Approximation Questions Discrete problem Discrete function spaces piecewise defined in a discrete domain (mesh) Classical & weak formulations? Properties of the fields? Finite element method Objective To build a discrete structure as similar as possible as the continuous structure 3

31 !! Discrete mathematical structure Finite element Interpolation in a geometric element of simple shape Finite element space Function space & Mesh Sequence of finite element spaces Sequence of function spaces & Mesh + f! + f i +! f i i i 31

32 Finite elements Finite element (K, P K, Σ K ) K = domain of space (tetrahedron, hexahedron, prism) P K = function space of finite dimension n K, defined in K Σ K = set of n K degrees of freedom represented by n K linear functionals φ i, 1 i n K, defined in P K and whose values belong to IR K = dom(f) x κ f P K f(x) cod(f) IR φ (f) i 32

33 Finite elements Unisolvance u P K, u is uniquely defined by the degrees of freedom Interpolation u K = n K i = 1 φ i ( u ) p i Degrees of freedom Basis functions Finite element space Union of finite elements (K j, P Kj, Σ Kj ) such as : the union of the K j fill the studied domain ( mesh) some continuity conditions are satisfied across the element interfaces 33

34 Sequence of finite element spaces Tetrahedral (4 nodes) Geometric elements Hexahedra (8 nodes) Prisms (6 nodes) Mesh Nodes i N Edges i E Faces i F Geometric entities Volumes i V S S 1 S 2 S 3 Sequence of function spaces 34

35 Sequence of finite element spaces Functions Properties Functionals Degrees of freedom S S 1 S 2 S 3 {s i, i N} {s i, i E} {s i, i F} {s i, i V} j j j s i (x j ) = δ ij i, j N s i. dl = δ ij s i. n ds = δ ij s i dv i, j E i, j F = δ ij i, j V Point evaluation Curve integral Surface integral Volume integral Nodal value Circulation along edge Flux across face Volume integral Nodal element Edge element Face element Volume element Bases u K = i φ i (u) s i Finite elements 35

36 Sequence of finite element spaces Base functions Continuity across element interfaces Codomains of the operators S S 1 S 2 S 3 {s i, i N} value {s i, i E} tangential component grad S S 1 {s i, i F} normal component curl S 1 S 2 {s i, i V} discontinuity div S 2 S 3 S S 1 grad S S 2 curl S 1 S 3 div S 2 Conformity Sequence S grad S 1 curl S 2 div S 3 36

37 Function spaces S et S 3 For each node i N scalar field s i (x) = p i (x) S p i = 1 at node i at all other nodes node i p i = 1 p i continuous in Ω For each Volume v V scalar field s v = 1 / vol (v) S 3 37

38 Edge function space S 1 For each edge e ij = {i, j} E vector field s e ij = p j grad p r - p i grad p r r N F, j i r N F, i j s e S 1 Definition of the set of nodes N F,mn - N F,mn = {i N ; i f mop (q), o, p,q n} p q p i Illustration of the vector field s e e ij s e j m o m o n n N F, ij - N F, jī N.B.: In an element : 3 edges/node 38

39 Edge function space S 1 i e ij j Geometric interpretation of the vector field s e p j grad p r r N F, j i N F, jī N s e F, j i ij = p j grad p r - p i grad p r r N F, j i r N F, i j i e ij j i e ij s e j N F, ij - r N F, i j - p i grad p r N F, i j N F, ij - N F, jī 39

40 Function space S 2 For each facet f F vector field f = f ijk(l) = {i, j, k (, l) } = {q 1, q 2, q 3 (, q 4 ) } s f = a f # N f p q c grad p r grad c = 1 r N F, q c q c + 1 r N F, q c q c - 1 p r s f S 2 #N f = 3 a f = 2 4 a f = 1 N F, ij - Illustration of the vector field s f i k N F, kj - f ijk N F, kī N F, ik - j N F, jk - N F, jī 4

41 Particular subspaces of S 1 Kernel of the curl operator Applications h S 1 (Ω) ; curl h = in Ω c C Ω h? Gauged subspace a S 1 (Ω) ; b = curl a S 2 (Ω) a? Gauge a. ω = to fix a Ampere equation in a domain Ω c C without current ( Ω c ) Gauge condition on a vector potential Definition of a generalized source field h s such that curl h s = j s 41

42 Kernel of the curl operator Case of simply connected domains Ω c C H = { h S 1 ( Ω ) ; curl h = in Ω c C } h = h a s a = h k s k e E h l = + h l s l C k E c l E c h. dl = - grad φ. dl = φ a l - φ b l l ab l ab h = grad φ in Ω c C C C N c, E c E c C C N c, E c (interface) Ω c Ω c h = h k s k + ( φ a l - φ b l ) s l k E c l E c C φ n' h k h = h k s k + φ n v n k E c with n N C c v n = s nj nj E C c φ n Base of H basis functions of inner edges of Ω c nodes of Ω cc, with those of Ω c 42

43 Kernel of the curl operator Case of multiply connected domains H = { h S 1 ( Ω ) ; curl h = in Ω c C } φ = φ cont + φ disc h = grad φ in Ω c C (cuts) φ + φ = φ Γ + eci φ Γ eci = I i φ disc = h = i C discontinuity of φ disc h k s k + φ cont n v n + I i c i k A c I i q i n N c C with edges of Ω c C starting from a node of the cut and located on side '+' but not on the cut c i = i C nj A C c n N ec i C j N c + j N ec i s nj q i defined in Ω c C unit discontinuity across Γ eci continuous in a transition layer zero out of this layer Basis of H basis functions of inner edges of Ω c nodes of Ω c C cuts of C 43

44 !! Gauged subspace of S 1 Gauged space in Ω b = curl a with a = a e s e S 1 ( Ω ), b = b f s f S 2 ( Ω ) e E f F b f = i ( e, f ) a e, f F matrix form: b f = C FE a e e E Tree set of edges connecting (in Ω) all the nodes of Ω without! forming any loop (E)! Co-tree complementary set of the tree (E) Face-edge incidence matrix tree! E Gauged space of S 1 (Ω)!! S 1 (Ω) = {a S 1 (Ω) ; a j =, j E}! a = a i s i S 1 ( Ω )! i E! Basis of S 1 (Ω) co-tree edge basis functions (explicit gauge definition)! co-tree E! 44

45 Mesh of electromagnetic devices Electromagnetic fields extend to infinity (unbounded domain) Approximate boundary conditions: zero fields at finite distance Rigorous boundary conditions: "infinite" finite elements (geometrical transformations) boundary elements (FEM-BEM coupling) Electromagnetic fields are confined (bounded domain) Rigorous boundary conditions 45

46 Mesh of electromagnetic devices Electromagnetic fields enter the materials up to a distance depending of physical characteristics and constraints Skin depth δ (δ<< if ω, σ, µ >>) δ = 2 ω σ µ mesh fine enough near surfaces (material boundaries) use of surface elements when δ 46

47 Mesh of electromagnetic devices Types of elements 2D : triangles, quadrangles 3D : tetrahedra, hexahedra, prisms, pyramids Coupling of volume and surface elements boundary conditions thin plates interfaces between regions cuts (for making domains simply connected) Special elements (air gaps between moving pieces,...) 47

48 Constraints in partial differential problems Local constraints (on local fields) Boundary conditions i.e., conditions on local fields on the boundary of the studied domain Interface conditions e.g., coupling of fields between sub-domains Global constraints (functional on fields) Flux or circulations of fields to be fixed e.g., current, voltage, m.m.f., charge, etc. Flux or circulations of fields to be connected e.g., circuit coupling Weak formulations for finite element models Essential and natural constraints, i.e., strongly and weakly satisfied 48

49 Constraints in electromagnetic systems Coupling of scalar potentials with vector fields e.g., in h-φ and a-v formulations Gauge condition on vector potentials e.g., magnetic vector potential a, source magnetic field h s Coupling between source and reaction fields e.g., source magnetic field h s in the h-φ formulation, source electric scalar potential v s in the a-v formulation Coupling of local and global quantities e.g., currents and voltages in h-φ and a-v formulations (massive, stranded and foil inductors) Interface conditions on thin regions i.e., discontinuities of either tangential or normal components Interest for a correct discrete form of these constraints Sequence of finite element spaces 49

50 Complementary 3D formulations Magnetodynamic h-formulation ( µ h, h' ) + ( σ 1 curlh, curlh' ) + < n e, h' > = h' F 1 h ( Ω) t Ω Ω s Γ e Magnetodynamic a-formulation ( µ 1 curl a, curl a' ) Ω + ( σ ta, a' ) Ω+ < n hs, a' > Γ = a' F1( ) h e Ω How to enforce global fluxes? 5

51 h-φ formulation h-φ magnetodynamic finite element formulations with massive and stranded inductors Use of edge and nodal finite elements for h and φ Natural coupling between h and φ Definition of current in a strong sense with basis functions either for massive or stranded inductors Definition of voltage in a weak sense Natural coupling between fields, currents and voltages etc. 51

52 a-v formulation a-v s Magnetodynamic finite element formulation with massive and stranded inductors Use of edge and nodal finite elements for a and v s Definition of a source electric scalar potential v s in massive inductors in an efficient way (limited support) Natural coupling between a and v s for massive inductors Adaptation for stranded inductors: several methods Natural coupling between local and global quantities, i.e. fields and currents and voltages etc. 52

53 Strong and weak formulations Equations curl h = j s div b = ρ s in Ω Strongly satisfies Scalar potential φ h = h s - grad φ, with curl h s = j s φ = constant each non-connected portion of Γ h Constitutive relation b = µ h Vector potential a b = b s + curl a, with div b s = ρ s n h = n h, n b = n b Γ h Boundary conditions (BCs) s Γ b s n a = n gradu Γ b Γ b 53

54 Strong formulations Electrokinetics curl e =, div j =, j = σ e, e = - grad v or j = curl u Electrostatics curl e =, div d = ρ s, d = ε e, e = - grad v or d = d s + curl u Magnetostatics curl h = j s, div b =, b = µ h, h = h s - grad φ or b = curl a Magnetodynamics curl h = j, curl e = t b, div b =, b = µ h, j = σ e + j s,... 54

55 Grad-div weak formulation grad-div Green formula ( u,grad v) + (div u, v) =< nu, v > Ω Ω 1 1 u grad v+ vdiv u= div( vu), u H ( Ω), v H ( Ω) Ω add on both sides: ρ (, φ ') +< nb, φ ' > s integration in Ω and divergence theorem Ω s, ρ s Γ b ( b,g rad φ ') ( ρ, φ') +< nb, φ ' > +< nb, φ ' > = R = Ω s Ωs, ρ s Γb Γφ b, φ' b=µ ( h grad φ) s = (div b ρ, φ ') < n b nb, φ' > s Ω s Γ b grad-div scalar potential φ weak formulation Magnetostatics: ρ s = Electrokinetics: ( j, grad v') +< n j, v' > +< n j, v' > = (div j, v') < n j n j, v' > = R Ω s Γ Γ Ω s Γ j, v' j v j Electrostatics: ( d, grad v') ( ρ, v') +< n d, v' > +< n d, v' > = (div d ρ, v') < n d n d, v' > = R Ω Ω s Γ Γ Ω s Γ d, v' s d v d 55

56 Curl-curl weak formulation curl-curl Green formula ( u,curl v ) Ω (curl u, v ) Ω = < n u, v> u curlv v curl u= div( v u), u, v H ( Ω) Ω add on both sides : ( j, a') +< n h, a' > s Ω s, j integration in Ω and divergence theorem s Γ h 1 ( h, curl a') Ω ( js, a') Ω +< n h, a' > +< n h, a' > = R =, s Γh Γb h, a' s j h=µ 1 ( b + curl a) s = (curl h j, a') < n h n h, a' > s Ω s Γ h curl-curl vector potential a weak formulation 56

57 Grad-div weak formulation Use of hierarchal TF φ p ' in the weak formulation 1 ( b,grad φ ') ( ρ, φ ') + < nb, φ ' > = R p Ω s p Ω s p Γ b φ s, ρ b, p ' Error indicator: lack of fulfillment of WF... can be used as a source for a local FE problem (naturally limited to the FE support of each TF) to calculate the higher order correction b p to be given to the actual solution b for satisfying the WF... solution of : ( b + b,grad φ ') ( ρ, φ ' ) +< n b, φ ' > = p p Ω s p Ω s p Γ s, ρ b or ( b,grad φ ') = R p p Ω b, φ ' p 1 Local FE problems 57

58 A posteriori error estimation (1/2) Electric scalar potential v (1st order) Coarse mesh Fine mesh Electrokinetic / electrostatic problem Electric field V Higher order hierarchal correction v p (2nd order, BFs and TFs on edges) Large local correction Large error Field discontinuity directly 58

59 Curl-curl weak formulation Use of hierarchal TF a p ' in the weak formulation 2 ( h,curl a ) (, '), a p' Ω js ap Ω +< n h ' s, j s p > Γ = R h h, ap ' Error indicator: lack of fulfillment of WF... also used as a source to calculate the higher order correction h p of h... solution of : Local FE problems 2 ( h,curl a ') p p Ω = R h, a p ' 59

60 A posteriori error estimation (2/2) V Magnetostatic problem Magnetodynamic problem Magnetic vector potential a (1st order) Fine mesh skin depth Higher order hierarchal correction a p (2nd order, BFs and TFs on faces) Magnetic core Coarse mesh Large local correction Large error Conductive core 6